Number Series 5, 8, 14, 26, 50: Doubling Versus Double-Minus-Two
A quantitative-reasoning item presents the number series 5, 8, 14, 26, 50, ___. A student, Andre, writes 100 and explains, 'Each number is about double the one before, so 50 times 2 is 100.' The teacher checks the transitions: 5 to 8 is not double, 8 to 14 is not double, and 14 to 26 is not double. The answer key gives 98. A second student, Bea, writes 98 and says, 'The differences are 3, 6, 12, 24, so the next difference is 48, and 50 plus 48 is 98.' The teacher wonders whether Bea's difference rule and the key's rule are actually the same, whether Andre's response is a minor arithmetic slip or a rule-construction error, and how to respond so that Andre learns to test a rule against every term rather than only the last one. The class has been working on both additive and multiplicative patterns.
1. Find a rule that generates 5, 8, 14, 26, 50 and determine the next term. Show that Bea's difference rule is equivalent.
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Rule: multiply by 2, then subtract 2. 5×2−2=8; 8×2−2=14; 14×2−2=26; 26×2−2=50; 50×2−2=98. Bea's difference rule gives differences 3, 6, 12, 24, then 48; 50+48=98. The rules are equivalent here: if aₙ = 2aₙ₋₁ − 2, then dₙ = aₙ − aₙ₋₁ = aₙ₋₁ − 2, and dₙ₊₁ = aₙ − 2 = 2aₙ₋₁ − 4 = 2(aₙ₋₁ − 2) = 2dₙ. So the differences double. Both students with 98 have a valid rule.
2. Is Andre's 100 an arithmetic slip or a rule-construction error? Support your judgment with evidence from the series.
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Rule-construction error. The arithmetic 50×2=100 is correct, but 'double' is the wrong operation. Check every transition: 5→8 is +3, not ×2; 8→14 is +6; 14→26 is +12. Only the last term is close to a double. His phrase 'about double' shows he noticed the rule did not fit exactly but did not revise. A true slip would be applying the correct rule and making a calculation error, such as writing 102 for 50×2−2. He never applied the −2, so the error is in rule selection.
3. Design a follow-up that helps Andre test his rule against every term and distinguish 'double' from 'double minus two.'
Read the worked answer
Have Andre build a three-column table: term, prediction from his 'double' rule, actual next term. Starting at 5, his rule predicts 10, but the series says 8. The conflict is immediate. Ask him to find an operation that turns 5 into 8 and also turns 8 into 14; guide him to 'double, then subtract 2.' Contrast with a pure doubling series (5, 10, 20, 40) so he can see the difference. Require him to state the rule and check it against every adjacent pair before writing the final answer.